The Lagrangian Method

What exactly is the Lagrangian method? It seems to be a popular method to solve Max/Min problems in Calculus. But generations of Calculus students may have found it troubling to understand why it works. We shall discuss this method today.

This is a method of finding local maxima and minima. Clearly, derivatives don’t tell us much about the global property of a function. They’re very much a local property. We’re supposed to maximize $f(x,y)$ under the condition that $g(x,y)=c$ . Note that $f(x,y)$ is embedded in three dimensions, while $g(x,y)=c$ is embedded in two.

In order to crack this problem, we need to rely upon the intuition that $f(x,y)$ at a critical point cannot increase/decrease anymore locally in the direction of the contour. The gradient is the direction along which a function sees its fastest increase/decrease. Hence, the direction in which it will increase/decrease lies completely orthogonal to the contour, which is exactly the direction in which the gradient of the contour lies.

Hence, $\nabla f=\lambda \nabla g$ .